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Full text of "Treatise On Analysis Vol-Ii"

8    COMPLETE HUBERT ALGEBRAS        363

In view of (15.8.6) and (15.8.3), we obtain the following corollary:

(15.8.8)    Every left ideal in A contains a minimal left ideal. Every minimal left
ideal is closed.

(15.8.9)    (i)   If e, e' are two orthogonal self-adjoint idempotents, then the left
ideals Ae and Ae' are orthogonal.

(ii)   Let (e^i^i^n be a finite family of pairwise orthogonal, self-adjoint

idempotents. Then for each x e A, the element x  ]T xe-t is orthogonal to each

i-l

(i)   We have (xe\ye*) = (xe\ye'2) = (xee'\yef) = 0, by (15.7.5.5)  and
(15.8.4).

(ii)   Likewise,

(n                \         /               n                     \

* - E ***Iyej] = (xej - ExeiejIyejj = 

for all/

(15.8.10)    For each XE A, there exists a finite or infinite sequence (en) of pair-
wise orthogonal, irreducible self-adoint idempotents belonging to the closure I of
the ideal Ax, such that x = ]T xen (this series being convergent in A), and

We may assume that x ^ 0. Then there exists at least one self-adjoint
idempotent e  I such that xe ^ 0, for the construction in (15.8.5) produces
a self-adjoint idempotent e ^ 0 in I such that e(x*x)2 = (?, and therefore
ex* = 0, so that xe = (ex*)* ^ 0. Since e is the sum of a finite number of
irreducible self-adjoint idempotents belonging to Ae (15.8.6), there exists at
least one of these idempotents whose product with x is ^0. We next remark
that if (ej)i $ i^n is a finite sequence of pairwise orthogonal, self-adjoint idem-
potents such that \\xet\\2 ^ a > 0 for all z, then by virtue of (15.8.9) we have

i=l        ||       i=l               1=1

and therefore n gjj ||x||2/a. Now define inductively an increasing sequence
(<p())n^0 of integers ^0; a (finite or infinite) sequence (ek) of pairwise
orthogonal, irreducible self-adjoint idempotents belonging to I; and a e2(e - e2) = 0,